A new approach to the representation theory of the partition category

Abstract

We explain a new approach to the representation theory of the partition category based on a reformulation of the definition of the Jucys-Murphy elements introduced originally by Halverson and Ram and developed further by Enyang. Our reformulation involves a new graphical monoidal category, the affine partition category, which is defined here as a certain monoidal subcategory of Khovanov's Heisenberg category. We use the Jucys-Murphy elements to construct some special projective functors, then apply these functors to give self-contained proofs of results of Comes and Ostrik on blocks of Deligne's category Rep(S_t).

References in corpus (13)

• Stability patterns in representation theory
• On blocks of Deligne's category Rep(S_t)
• The stability of the Kronecker products of Schur functions
• Trace decategorification of the Hecke category
• A seminormal form for partition algebras
• Symmetric group characters as symmetric functions
• The representation theory of Brauer categories I: triangular categories
• Jucys-Murphy elements and a presentation for partition algebras
• Heisenberg and Kac-Moody categorification
• Semisimplification of the category of tilting modules for GL_n
• The Center of the Partition Algebra
• Deligne categories and reduced Kronecker coefficients
• Embedding Deligne's category $\underline{\mathrm{Re}}\mathrm{p}(S_t)$ in the Heisenberg category